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Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series

Year 2023, , 130 - 139, 18.12.2023
https://doi.org/10.32323/ujma.1359248

Abstract

Let $p\in {\Bbb N}$, $s=(s_1,\ldots,s_p)\in {\Bbb C}^p$, $h=(h_1,\ldots,h_p)\in {\Bbb R}^p_+$, $(n)=(n_1,\ldots,n_p)\in {\Bbb N}^p$ and the sequences $\lambda_{(n)}=(\lambda^{(1)}_{n_1},\ldots,\lambda^{(p)}_{n_p})$ are such that $0<\lambda^{(j)}_1<\lambda^{(j)}_k<\lambda^{(j)}_{k+1}\uparrow+\infty$
as $k\to\infty$ for every $j=1,\ldots,p$. For $a=(a_1,\ldots,a_p)$ and $c=(c_1,\ldots,c_p)$ let $(a,c)=a_1c_1+\ldots+a_pc_p$, and we say that $a>c$ if $a_j> c_j$ for all $1\le j\le p$. For a multiple Dirichlet series \begin{align*}F(s)=e^{(s,h)}+\sum\limits_{\lambda_{(n)}>h}f_{(n)}\exp\{(\lambda_{(n)},s)\}\end{align*} absolutely converges in $\Pi^p_0=\{s:\text{Re}\,s<0\}$, concepts of pseudostarlikeness and pseudoconvexity are introduced and criteria for pseudostarlikeness and the pseudoconvexity are proved. Using the obtained results, we investigated neighborhoods of multiple Dirichlet series, Hadamard compositions, and properties of solutions of some differential equations.

References

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  • [2] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8(3) (1957), 597–601.
  • [3] M. M.Sheremeta, Geometric Properties of Analytic Solutions of Differential Equations, Lviv: Publisher I. E. Chyzhykov, 2019.
  • [4] I. S. Jack, Functions starlike and convex of order a, J. London Math. Soc., 3 (1971), 469–474.
  • [5] V. P. Gupta, Convex class of starlike functions, Yokohama Math. J., 32 (1984), 55–59.
  • [6] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8 (1957), 598–601.
  • [7] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81(4) (1981), 521–527.
  • [8] R. Fournier, A note on neighborhoods of univalent functions, Proc. Amer. Math. Soc., 87(1) (1983), 117–121.
  • [9] H. Silverman, Neighborhoods of a class of analytic functions, Far East J. Math. Sci., 3(2) (1995), 165–169.
  • [10] O. Altıntas¸, Neighborhoods of certain analytic functions with negative coefficients, Int. J. Math. and Math. Sci. 13(4) (1996), 210–219.
  • [11] O. Altıntaş, O. Özkan, H. M.Srivastava, Neighborhoods of a class of analytic functions with negative coefficients, Applied Math. Lettr., 13 (2000), 63–67.
  • [12] B. A. Frasin, M. Daras, Integral means and neighborhoods for analytic functions with negative coefficients, Soochow Journal Math., 30(2) (2004), 217–223.
  • [13] G. Murugusundaramoorthy, H. M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure and Appl. Math., 5(2) (2004), Article 24.
  • [14] M. N. Pascu, N. R. Pascu, Neighborhoods of univalent functions, Bull. Amer. Math. Soc., 83 (2011), 510–219.
  • [15] J. Hadamard, Theoreme sur le series entieres, Acta Math., 22 (1899), 55–63.
  • [16] J. Hadamard, La serie de Taylor et son prolongement analitique, Scientia Phys. Math., 12 (1901), 43–62.
  • [17] L. Bieberbach, Analytische Fortzetzung, Berlin, 1955.
  • [18] Yu. F. Korobeinik, N. N. Mavrodi, Singular points of the Hadamard composition, Ukr. Math. J., 42(12) (1990), 1711–1713. (in Russian); Engl. Transl.: Ukr. Math. J., 42(12) (1990), 1545–1547.
  • [19] L. Zalzman, Hadamard product of shlicht functions, Proc. Amer. Math. Soc., 19(3) (1968), 544–548.
  • [20] M. L. Mogra, Hadamard product of certain meromorphic univalent functions, J. Math. Anal. Appl., 157 (1991), 10–16.
  • [21] J. H. Choi, Y. C. Kim, S. Owa, Generalizations of Hadamard products of functions with negative coefficients, J. Math. Anal. Appl., 199 (1996), 495–501.
  • [22] M. K. Aouf, H. Silverman, Generalizations of Hadamard products of meromorphic univalent functions with positive coefficients, Demonstr. Math. 51(2) (2008), 381–388.
  • [23] J. Liu, P. Srivastava, Hadamard products of certain classes of p-valent starlike functions, RACSM. 113 (2019), 2001–205.
  • [24] O. M. Holovata, O. M. Mulyava, M. M. Sheremeta, Pseudostarlike, pseudoconvex and close-to pseudoconvex Dirichlet series satisfying differential equations with exponential coefficients, Mat. Methods and Fiz.-Mech. Polya., 61(1) (2018), 57–70.
  • [25] M. M. Sheremeta, Pseudostarlike and pseudoconvex Dirichlet series of the order a and the type b, Mat. Stud. 54(1) (2020), 23–31.
  • [26] S. M. Shah, Univalence of a function f and its successive derivatives when f satisfies a differential equation, II, J. Math. Anal. and Appl., 142 (1989), 422–430.
  • [27] Z. M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14(1) (2000), 54–58.
  • [28] Z. M. Sheremeta, M. M. Sheremeta, Convexity of entire solutions of a differential equation, Mat. Methods and Fiz.-Mech. Polya., 47(2) (2004), 181–185.
  • [29] M. M. Sheremeta, Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients, Mat. Stud., 56(1) (2021), 39–47.
  • [30] M. M. Sheremeta, On certain subclass of Dirichlet series absolutely convergent in half-plane, Mat. Stud., 57(1)(2022), 32–44.
  • [31] M. M. Sheremeta, O. B. Skaskiv, Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series, Mat. Stud., 58(2) (2023), 182–200.
Year 2023, , 130 - 139, 18.12.2023
https://doi.org/10.32323/ujma.1359248

Abstract

References

  • [1] G. M. Golusin, Geometrical theory of functions of complex variables, M. Nauka, 1966. (in Russian); Engl. Transl.: AMS: Translations of Mathematical Monograph, 26 (1969).
  • [2] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8(3) (1957), 597–601.
  • [3] M. M.Sheremeta, Geometric Properties of Analytic Solutions of Differential Equations, Lviv: Publisher I. E. Chyzhykov, 2019.
  • [4] I. S. Jack, Functions starlike and convex of order a, J. London Math. Soc., 3 (1971), 469–474.
  • [5] V. P. Gupta, Convex class of starlike functions, Yokohama Math. J., 32 (1984), 55–59.
  • [6] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8 (1957), 598–601.
  • [7] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81(4) (1981), 521–527.
  • [8] R. Fournier, A note on neighborhoods of univalent functions, Proc. Amer. Math. Soc., 87(1) (1983), 117–121.
  • [9] H. Silverman, Neighborhoods of a class of analytic functions, Far East J. Math. Sci., 3(2) (1995), 165–169.
  • [10] O. Altıntas¸, Neighborhoods of certain analytic functions with negative coefficients, Int. J. Math. and Math. Sci. 13(4) (1996), 210–219.
  • [11] O. Altıntaş, O. Özkan, H. M.Srivastava, Neighborhoods of a class of analytic functions with negative coefficients, Applied Math. Lettr., 13 (2000), 63–67.
  • [12] B. A. Frasin, M. Daras, Integral means and neighborhoods for analytic functions with negative coefficients, Soochow Journal Math., 30(2) (2004), 217–223.
  • [13] G. Murugusundaramoorthy, H. M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure and Appl. Math., 5(2) (2004), Article 24.
  • [14] M. N. Pascu, N. R. Pascu, Neighborhoods of univalent functions, Bull. Amer. Math. Soc., 83 (2011), 510–219.
  • [15] J. Hadamard, Theoreme sur le series entieres, Acta Math., 22 (1899), 55–63.
  • [16] J. Hadamard, La serie de Taylor et son prolongement analitique, Scientia Phys. Math., 12 (1901), 43–62.
  • [17] L. Bieberbach, Analytische Fortzetzung, Berlin, 1955.
  • [18] Yu. F. Korobeinik, N. N. Mavrodi, Singular points of the Hadamard composition, Ukr. Math. J., 42(12) (1990), 1711–1713. (in Russian); Engl. Transl.: Ukr. Math. J., 42(12) (1990), 1545–1547.
  • [19] L. Zalzman, Hadamard product of shlicht functions, Proc. Amer. Math. Soc., 19(3) (1968), 544–548.
  • [20] M. L. Mogra, Hadamard product of certain meromorphic univalent functions, J. Math. Anal. Appl., 157 (1991), 10–16.
  • [21] J. H. Choi, Y. C. Kim, S. Owa, Generalizations of Hadamard products of functions with negative coefficients, J. Math. Anal. Appl., 199 (1996), 495–501.
  • [22] M. K. Aouf, H. Silverman, Generalizations of Hadamard products of meromorphic univalent functions with positive coefficients, Demonstr. Math. 51(2) (2008), 381–388.
  • [23] J. Liu, P. Srivastava, Hadamard products of certain classes of p-valent starlike functions, RACSM. 113 (2019), 2001–205.
  • [24] O. M. Holovata, O. M. Mulyava, M. M. Sheremeta, Pseudostarlike, pseudoconvex and close-to pseudoconvex Dirichlet series satisfying differential equations with exponential coefficients, Mat. Methods and Fiz.-Mech. Polya., 61(1) (2018), 57–70.
  • [25] M. M. Sheremeta, Pseudostarlike and pseudoconvex Dirichlet series of the order a and the type b, Mat. Stud. 54(1) (2020), 23–31.
  • [26] S. M. Shah, Univalence of a function f and its successive derivatives when f satisfies a differential equation, II, J. Math. Anal. and Appl., 142 (1989), 422–430.
  • [27] Z. M. Sheremeta, On entire solutions of a differential equation, Mat. Stud., 14(1) (2000), 54–58.
  • [28] Z. M. Sheremeta, M. M. Sheremeta, Convexity of entire solutions of a differential equation, Mat. Methods and Fiz.-Mech. Polya., 47(2) (2004), 181–185.
  • [29] M. M. Sheremeta, Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients, Mat. Stud., 56(1) (2021), 39–47.
  • [30] M. M. Sheremeta, On certain subclass of Dirichlet series absolutely convergent in half-plane, Mat. Stud., 57(1)(2022), 32–44.
  • [31] M. M. Sheremeta, O. B. Skaskiv, Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series, Mat. Stud., 58(2) (2023), 182–200.
There are 31 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Articles
Authors

Myroslav Sheremeta 0000-0002-8691-7463

Early Pub Date November 22, 2023
Publication Date December 18, 2023
Submission Date September 12, 2023
Acceptance Date October 29, 2023
Published in Issue Year 2023

Cite

APA Sheremeta, M. (2023). Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series. Universal Journal of Mathematics and Applications, 6(4), 130-139. https://doi.org/10.32323/ujma.1359248
AMA Sheremeta M. Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series. Univ. J. Math. Appl. December 2023;6(4):130-139. doi:10.32323/ujma.1359248
Chicago Sheremeta, Myroslav. “Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series”. Universal Journal of Mathematics and Applications 6, no. 4 (December 2023): 130-39. https://doi.org/10.32323/ujma.1359248.
EndNote Sheremeta M (December 1, 2023) Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series. Universal Journal of Mathematics and Applications 6 4 130–139.
IEEE M. Sheremeta, “Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series”, Univ. J. Math. Appl., vol. 6, no. 4, pp. 130–139, 2023, doi: 10.32323/ujma.1359248.
ISNAD Sheremeta, Myroslav. “Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series”. Universal Journal of Mathematics and Applications 6/4 (December 2023), 130-139. https://doi.org/10.32323/ujma.1359248.
JAMA Sheremeta M. Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series. Univ. J. Math. Appl. 2023;6:130–139.
MLA Sheremeta, Myroslav. “Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series”. Universal Journal of Mathematics and Applications, vol. 6, no. 4, 2023, pp. 130-9, doi:10.32323/ujma.1359248.
Vancouver Sheremeta M. Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series. Univ. J. Math. Appl. 2023;6(4):130-9.

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